Spherical harmonics and infinitesimal projective transformations

Consider the space of $C^{\mathrm{\infty }}$-functions on the two-dimensional sphere $S^2$ and its decomposition $\oplus {\mathcal{H}}_m$ into a direct sum of minimal rotation-invariant spaces. We consider elements of $\oplus {\mathcal{H}}_m$ as functions of two variables, a nonnegative integer variable $m$ and a complex variable $u$ (a restriction of such function to a set $m=k$ is a polynomial in $u$ of degree $\le 2k$). The group $\mathrm{G}\mathrm{L}(3,\mathbb{R})$ of all matrices of order 3 acts on $S^2$, so its Lie algebra acts in $C^{\mathrm{\infty }}(S^2)$ by differentiations along vector fields. We construct operators in ${\mathcal{W}}^{\mathrm{\infty }}$ corresponding such differentiations and obtain a collection of differential-difference operators in ${\mathcal{W}}^{\mathrm{\infty }}$. The expressions include shifts $m\mapsto m\pm 2$ and derivatives of order 4 in $u$.